论文标题
Hopf形式和Hopf-Galois理论
Hopf Forms and Hopf-Galois Theory
论文作者
论文摘要
令$ k $为$ \ q $的有限字段扩展名，让$ n $为有限组，具有自动形态组$ f = \ aut（n）$。 R.Haggenmüller和B. pareigis表明，从$ f $ f $ -galois的$ galois扩展$ k $ of $ k $ of $ k $ of hopf $ k $ hopf $ k [n]中，有一个两次两次两次两者。对于$ n = c_n $，$ n \ ge 1 $，$ c_p^m $，$ p $ prime，$ m \ ge 1 $和$ n = d_3，d_4，q_4，q_8 $，我们表明$ \ q [n] $承认绝对是半的semisimple hopf form $ h $ $ h $，并找到$ l $ for $ l $ for $ l $ for $ f $ f $ umumuth $ fum- umumuth $ fum- umuth $ fum- umuth $ fum- umuth $ fum- $ funs = h $ = h $ = h $ = h $。此外，如果$ h $是Hopf-Galois结构在Galois扩展名$ E/K $上给出的HOPF代数，我们将展示如何在$θ$下构建$ h $的预先映射，并在$θ$以下。
Let $K$ be a finite field extension of $\Q$ and let $N$ be a finite group with automorphism group $F=\Aut(N)$. R. Haggenmüller and B. Pareigis have shown that there is a bijection \[Θ: {\mathcal Gal}(K,F)\rightarrow {\mathcal Hopf}(K[N])\] from the collection of $F$-Galois extensions of $K$ to the collection of Hopf forms of the group ring $K[N]$. For $N=C_n$, $n\ge 1$, $C_p^m$, $p$ prime, $m\ge 1$, and $N=D_3,D_4,Q_8$, we show that $\Q[N]$ admits an absolutely semisimple Hopf form $H$ and find $L$ for which $Θ(L)=H$. Moreover, if $H$ is the Hopf algebra given by a Hopf-Galois structure on a Galois extension $E/K$, we show how to construct the preimage of $H$ under $Θ$ assuming certain conditions.
